﻿-?=P„T°{-T 
  +2 
  -£ 
  I 
  + 
  te 
  - 
  } 
  -... 
  (105.) 
  

  

  MECHANICAL 
  ACTION 
  OF 
  HEAT. 
  581 
  

  

  P 
  V 
  V 
  P 
  

  

  =? 
  — 
  -5 
  • 
  hyp. 
  log 
  ^ 
  = 
  N 
  • 
  hyp. 
  log 
  ^ 
  nearly 
  (t 
  being 
  nearly 
  constant), 
  

  

  and 
  K 
  v 
  nearly 
  = 
  ft. 
  

  

  The 
  value 
  of 
  the 
  integral 
  in 
  the 
  numerator 
  is 
  found 
  as 
  follows 
  : 
  — 
  

   The 
  Centrifugal 
  Theory 
  of 
  Elasticity 
  indicates 
  that 
  the 
  pressure 
  of 
  an 
  imper- 
  

   fect 
  gas 
  may 
  be 
  represented 
  by 
  the 
  following 
  formula: 
  — 
  

  

  P=P 
  ^(^+A 
  -^-^-&c. 
  } 
  (104.) 
  

  

  where 
  V 
  is 
  the 
  volume 
  in 
  the 
  perfectly 
  gaseous 
  state, 
  at 
  a 
  standard 
  pressure 
  P 
  a 
  , 
  

   and 
  absolute 
  temperature 
  t 
  , 
  and 
  A 
  , 
  A 
  1? 
  &c, 
  are 
  a 
  series 
  of 
  functions 
  of 
  the 
  den- 
  

   sity, 
  to 
  be 
  determined 
  empirically. 
  From 
  this 
  formula 
  it 
  is 
  easily 
  seen 
  that 
  

   dP 
  

  

  d 
  t 
  

  

  so 
  that 
  the 
  first 
  term 
  in 
  the 
  numerator 
  of 
  the 
  expression 
  (103) 
  has 
  the 
  following 
  

   value 
  : 
  — 
  

  

  P 
  V 
  

   in 
  which 
  -j^— 
  2 
  = 
  Nt 
  nearly. 
  

  

  In 
  order 
  to 
  represent 
  correctly 
  the 
  result 
  of 
  M. 
  Regnault's 
  experiments 
  on 
  

   the 
  elasticity 
  and 
  expansion 
  of 
  gases, 
  it 
  was 
  found 
  sufficient 
  to 
  use, 
  in 
  the 
  for- 
  

   mula 
  for 
  the 
  pressure 
  (104), 
  the 
  first 
  three 
  terms 
  ; 
  and 
  the 
  functions 
  of 
  the 
  den- 
  

   sity 
  which 
  occur 
  in 
  these 
  terms, 
  as 
  determined 
  empirically 
  from 
  the 
  experiments, 
  

   were 
  found 
  to 
  have 
  the 
  following 
  values, 
  in 
  which 
  the 
  unit 
  of 
  volume 
  is 
  the 
  

   theoretical 
  volume 
  of 
  unity 
  of 
  weight 
  of 
  air 
  under 
  the 
  pressure 
  of 
  one 
  atmosphere, 
  

   at 
  the 
  temperature 
  of 
  melting 
  ice,* 
  and 
  the 
  values 
  of 
  the 
  constants 
  are 
  given 
  for 
  

   the 
  centigrade 
  scale. 
  

  

  V 
  =6 
  (^) 
  f; 
  V= 
  a 
  (vY 
  -■-.. 
  (107.) 
  

   Com. 
  log 
  b 
  = 
  3-8181545 
  ; 
  Com. 
  log 
  a 
  = 
  0-3176168. 
  

  

  Hence 
  it 
  appears 
  that 
  the 
  integrals 
  in 
  the 
  formula 
  (106) 
  have 
  the 
  following 
  

   values 
  : 
  — 
  

  

  y^v.„. 
  A: 
  (4).^>v.».±.*... 
  (i)* 
  .(iota.) 
  

  

  in 
  which 
  the 
  common 
  logarithms 
  of 
  the 
  constants 
  are 
  

  

  * 
  This 
  unit 
  of 
  volume 
  is 
  greater 
  than 
  the 
  actual 
  volume 
  of 
  air, 
  under 
  the 
  circumstances 
  described, 
  

   in 
  the 
  ratio 
  of 
  1 
  00085 
  to 
  1. 
  

  

  VOL. 
  XX. 
  PART 
  IV. 
  7 
  S 
  

  

  